Single-Context Visible and Remainder Dimensions

Abstract

One rank-one projective context exposes one diagonal slice of the trace-zero Hermitian state directions and leaves its orthogonal remainder.

Theorem 1.1 (A single context exposes exactly its diagonal share).

$$\begin{array}{c}\forall d,B:\mathrm{RankOneContext}\left(d\right),d\ge 2\wedge \mathrm{IsRecordMeasurement}\left(\mathrm{projector}\left(B\right)\right)\Rightarrow \\ \mathrm{finrankR}\left(\mathrm{diagonalTraceZeroSubspace}\left(B\right)\right)\le d-1\wedge \\ \mathrm{finrankR}\left(\mathrm{orthogonal}\left(\mathrm{diagonalTraceZeroSubspace}\left(B\right)\right)\right)=d^2-d\wedge \\ \mathrm{visibleRatio}\left(B\right)=\frac{d-1}{d^2-1}\wedge \\ \mathrm{visibleRatio}\left(B\right)=\frac{1}{d+1}\wedge \\ \mathrm{remainderRatio}\left(B\right)=\frac{d^2-d}{d^2-1}\wedge \\ \mathrm{remainderRatio}\left(B\right)=\frac{d}{d+1}\wedge \\ \mathrm{probabilityVectorExposedRatio}\left(B\right)=\frac{1}{d+1}.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/MeasurementGeometry/SingleContextVisibleRemainderDimension.single_context_visible_remainder_dimension (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let d be at least two and let B be a complete rank-one projective measurement. Its visible carrier is the repository’s canonical diagonalTraceZeroSubspace inside traceZeroHermitian d; the unread carrier is its real Hilbert–Schmidt orthogonal complement.

The visible finrank is at most d minus one, while the orthogonal remainder finrank is d squared minus d. Dividing each by the canonical trace-zero Hermitian finrank gives both displayed forms of the visible and remainder ratios.

The last clause uses the range of contextProbabilityDirection, the actual real-linear probability-vector readout on trace-zero state directions. Its ratio is one over d plus one. This is a ratio of linear dimensions, not probability mass of an individual state.

References

• Truth anchor: D5/S3/Quantum/MeasurementGeometry/SingleContextVisibleRemainderDimension.single_context_visible_remainder_dimension
• Dependency: D5/S3/Quantum/Entanglement/BipartiteSectorDecomposition
• Dependency: D5/S3/Quantum/Tomography/OneStepProbabilityInnovation